Difference between revisions of "Rayleigh sigma estimate"
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| − | == Unknown true center | + | == Unknown true center == |
| + | === Deriving Singh's (1992) <math>C_{2}</math> estimator === | ||
| − | When the <math>(\mu_{X}, \mu_{Y})</math>-center is estimated by <math>(\bar{x}, \bar{y})</math>, the Bessel correction is required to make the ML-estimate of the variance unbiased: | + | When the <math>(\mu_{X}, \mu_{Y})</math>-center is estimated by <math>(\bar{x}, \bar{y})</math>, the [[Closed_Form_Precision#Bessel_correction_factor|Bessel correction]] is required to make the ML-estimate of the variance unbiased: |
<math> | <math> | ||
| − | \widehat{\sigma^{2}_{ub}} = \frac{N}{N-1} \cdot \frac{1}{2N} \cdot \text{SSR} | + | \widehat{\sigma^{2}_{ub}} = \frac{N}{N-1} \cdot \frac{1}{2N} \cdot \text{SSR} = \frac{1}{2(N-1)} \cdot \text{SSR} |
</math> | </math> | ||
| − | However, taking the square root of the unbiased variance estimate makes it biased ([http://en.wikipedia.org/wiki/Jensen%27s_inequality Jensen's inequality]). Specifically, since the square root is concave, the bias is negative and <math>\sqrt{\widehat{\sigma^{2}_{ub}}}</math> underestimates <math>\sigma</math>. | + | However, taking the square root of the unbiased variance estimate makes it biased ([http://en.wikipedia.org/wiki/Jensen%27s_inequality Jensen's inequality]). Specifically, since the square root is concave, the bias is negative and <math>\sqrt{\widehat{\sigma^{2}_{ub}}}</math> underestimates <math>\sigma</math>. Another correction factor is needed to counter this effect. To this end, let <math>Q</math> be a <math>\chi</math>-distributed variable with <math>k</math> degrees of freedom. Then its mean is: |
<math> | <math> | ||
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</math> | </math> | ||
| − | When <math>X</math> is a normally distributed random variable with <math>n</math> observations, and <math>s^{2} := \frac{1}{n-1} \sum\limits_{i=1}^{n}(x-\bar{x})^{2}</math> is the Bessel-corrected variance estimate, <math>E(s^{2}) = \sigma^{2}</math>. Then <math>c_{4}(n)</math> is the correction factor such that <math>E(s) = c_{4}(n) \cdot \sigma</math>. With <math>Q</math> as given above, <math>c_{4}(n) = \frac{1}{\sqrt{k}} \cdot E(Q)</math> with <math>k := n-1</math>: | + | When <math>X</math> is a normally distributed random variable with <math>n</math> observations, and <math>s^{2} := \frac{1}{n-1} \sum\limits_{i=1}^{n}(x-\bar{x})^{2}</math> is the Bessel-corrected variance estimate, <math>E(s^{2}) = \sigma^{2}</math>. Then [[Closed_Form_Precision#Gaussian_correction_factor|<math>c_{4}(n)</math> is the correction factor]] such that <math>E(s) = c_{4}(n) \cdot \sigma</math>. With <math>Q</math> as given above, <math>c_{4}(n) = \frac{1}{\sqrt{k}} \cdot E(Q)</math> with <math>k := n-1</math>: |
<math> | <math> | ||
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</math> | </math> | ||
| − | + | We lose 2 degrees of freedom estimating the center. Therefore we really only have 2N - 2 degrees of freedom, so we set <math>n := 2N-1</math> so that <math>c_{4}(n)</math> is the scaled mean of a <math>\chi</math>-distributed variable with <math>k = n-1 = 2N-1-1 = 2N-2</math> degrees of freedom. This gives: | |
<math> | <math> | ||
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</math> | </math> | ||
| − | The | + | The penultimate expression is [[CEP_literature#Singh1992|Singh's 1992]] definition 3.3 for estimator <math>C_{2}</math>, also given by [[CEP_literature#Moranda1959|Moranda (1959)]]. |
| − | == Known true center | + | == Known true center == |
| + | === Deriving Singh's (1992) <math>C_{1}</math> estimator === | ||
| − | When the <math>(\mu_{X}, \mu_{Y})</math>-center is known, the Bessel correction is '''not''' required to make the Maximum-Likelihood-estimate of the variance unbiased. | + | When the <math>(\mu_{X}, \mu_{Y})</math>-center is known, the Bessel correction is '''not''' required to make the Maximum-Likelihood-estimate of the variance unbiased. In this case we simply set <math>n := 2N+1</math> for the <math>c_{4}(n)</math> correction factor so that <math>c_{4}(n)</math> is the scaled mean of a <math>\chi</math>-distributed variable with <math>k = n-1 = 2N+1-1 = 2N</math> degrees of freedom. This gives: |
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<math> | <math> | ||
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</math> | </math> | ||
| − | The | + | The penultimate expression is [[CEP_literature#Singh1992|Singh's 1992]] definition 2.2 for estimator <math>C_{1}</math>. |
Latest revision as of 11:27, 30 May 2017
Contents
Estimating parameter \(\sigma\) for the Rayleigh distribution
Let \(X, Y\) be uncorrelated, jointly normally distributed cartesian coordinates with means \(\mu_{X}, \mu_{Y}\), and equal variances \(\sigma_{X}^{2} = \sigma_{Y}^{2}\). The radius of an \((X,Y)\)-coordinate pair around the true mean is\[ R := \sqrt{(X-\mu_{X})^{2} + (Y-\mu_{Y})^{2}} \]
With \(N\) observations of \((x,y)\)-coordinates, \(\text{SSR}\) is the sum of squared radii\[ \begin{array}{rcl} \text{SSR} &:=& \sum\limits_{i=1}^{N} r_{i}^{2} = \sum\limits_{i=1}^{N} ((x_{i}-\mu_{X})^{2} + (y_{i}-\mu_{Y})^{2})\\ &=& \sum\limits_{i=1}^{N} (x_{i}-\mu_{X})^{2} + \sum\limits_{i=1}^{N} (y_{i}-\mu_{Y})^{2} \end{array} \]
The Maximum-Likelihood-estimate of the variance of \(r\) is the total variance of all \(2N\) separate \(x\)- and \(y\)-coordinates\[ \widehat{\sigma^{2}_{ML}} = \frac{1}{2N} \cdot \text{SSR} \]
Unknown true center
Deriving Singh's (1992) \(C_{2}\) estimator
When the \((\mu_{X}, \mu_{Y})\)-center is estimated by \((\bar{x}, \bar{y})\), the Bessel correction is required to make the ML-estimate of the variance unbiased\[ \widehat{\sigma^{2}_{ub}} = \frac{N}{N-1} \cdot \frac{1}{2N} \cdot \text{SSR} = \frac{1}{2(N-1)} \cdot \text{SSR} \]
However, taking the square root of the unbiased variance estimate makes it biased (Jensen's inequality). Specifically, since the square root is concave, the bias is negative and \(\sqrt{\widehat{\sigma^{2}_{ub}}}\) underestimates \(\sigma\). Another correction factor is needed to counter this effect. To this end, let \(Q\) be a \(\chi\)-distributed variable with \(k\) degrees of freedom. Then its mean is\[ E(Q) = \sqrt{2} \cdot \frac{\Gamma\left(\frac{k+1}{2}\right)}{\Gamma\left(\frac{k}{2}\right)} \]
When \(X\) is a normally distributed random variable with \(n\) observations, and \(s^{2} := \frac{1}{n-1} \sum\limits_{i=1}^{n}(x-\bar{x})^{2}\) is the Bessel-corrected variance estimate, \(E(s^{2}) = \sigma^{2}\). Then \(c_{4}(n)\) is the correction factor such that \(E(s) = c_{4}(n) \cdot \sigma\). With \(Q\) as given above, \(c_{4}(n) = \frac{1}{\sqrt{k}} \cdot E(Q)\) with \(k := n-1\)\[ c_{4}(n) = \sqrt{\frac{2}{n-1}} \cdot \frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)} \]
We lose 2 degrees of freedom estimating the center. Therefore we really only have 2N - 2 degrees of freedom, so we set \(n := 2N-1\) so that \(c_{4}(n)\) is the scaled mean of a \(\chi\)-distributed variable with \(k = n-1 = 2N-1-1 = 2N-2\) degrees of freedom. This gives\[ \begin{array}{rcl} \widehat{\sigma_{ub}} &=& \frac{1}{c_{4}(2N-1)} \cdot \sqrt{\widehat{\sigma^{2}_{ub}}} \\ &=& \frac{1}{\sqrt{\frac{2}{2N-1-1}} \cdot \frac{\Gamma\left(\frac{2N-1}{2}\right)}{\Gamma\left(\frac{2N-1-1}{2}\right)}} \cdot \sqrt{\frac{N}{N-1} \cdot \frac{1}{2N} \cdot \text{SSR}} \\ &=& \sqrt{\frac{2(N-1)}{2}} \cdot \frac{\Gamma\left(\frac{2(N-1)}{2}\right)}{\Gamma\left(\frac{2N-1}{2}\right)} \cdot \sqrt{\frac{N}{N-1} \cdot \frac{1}{2N} \cdot \text{SSR}} \\ &=& \sqrt{N} \cdot \frac{\Gamma(N-1)}{\Gamma\left(\frac{2N-1}{2}\right)} \cdot \sqrt{\frac{1}{2N} \cdot \text{SSR}} \\ &=& \frac{\Gamma(N-1)}{\Gamma\left(N - \frac{1}{2}\right)} \cdot \sqrt{\frac{1}{2} \cdot \text{SSR}} \end{array} \]
The penultimate expression is Singh's 1992 definition 3.3 for estimator \(C_{2}\), also given by Moranda (1959).
Known true center
Deriving Singh's (1992) \(C_{1}\) estimator
When the \((\mu_{X}, \mu_{Y})\)-center is known, the Bessel correction is not required to make the Maximum-Likelihood-estimate of the variance unbiased. In this case we simply set \(n := 2N+1\) for the \(c_{4}(n)\) correction factor so that \(c_{4}(n)\) is the scaled mean of a \(\chi\)-distributed variable with \(k = n-1 = 2N+1-1 = 2N\) degrees of freedom. This gives\[ \begin{array}{rcl} \widehat{\sigma_{ub}} &=& \frac{1}{c_{4}(2N+1)} \cdot \sqrt{\widehat{\sigma^{2}_{ub}}} \\ &=& \frac{1}{\sqrt{\frac{2}{2N+1-1}} \cdot \frac{\Gamma\left(\frac{2N+1}{2}\right)}{\Gamma\left(\frac{2N+1-1}{2}\right)}} \cdot \sqrt{\frac{1}{2N} \cdot \text{SSR}} \\ &=& \sqrt{N} \cdot \frac{\Gamma(N)}{\Gamma\left(\frac{2N+1}{2}\right)} \cdot \sqrt{\frac{1}{2N} \cdot \text{SSR}} \\ &=& \frac{\Gamma(N)}{\Gamma\left(N + \frac{1}{2}\right)} \cdot \sqrt{\frac{1}{2} \cdot \text{SSR}} \end{array} \]
The penultimate expression is Singh's 1992 definition 2.2 for estimator \(C_{1}\).
